The following documents will be mentioned in the description below:
Efros, A. A., Leung, T. K., 1999, Texture Synthesis by Non-Parametric Sampling, The Proceedings of the Seventh IEEE International Conference on Computer Vision, 2, 1033-1038, http://dx.doi.org/10.1109/ICCV.1999.790383,
Guardiano, F., Srivastava, M., 1993, Multivariate Geostatistics: Beyond Bivariate Moments, in: Soares, A. (Ed.), Geostatistics-Troia. Kluwer Academic Publications, Dordrecht, 133-144, http://dx.doi.org/10.1007/978-94-011-1739-5 12,
Müller, C., Siegesmund, S., Blum, P., 2010, Evaluation of the Representative Elementary Volume (REV) of a Fractured Geothermal Sandstone Reservoir, Environ. Earth Sci., 61, 1713-1724, http://dx.doi.org/10.1007/s.12665-010-0485-7,
Strebelle, S., 2002, Conditional Simulation of Complex Geological Structures Using Multiple-Point Statistics, Mathematical Geology, 34(1), 1-21,
Wei, L. Y., Levoy, M., 2000, Fast Texture Synthesis Using Tree-Structured Vector Quantization, SIGGRAPH'00, Proceedings of 27th Ann. Conf. Comp. Graphics Inter. Techn., 479-488, http://dx.doi.org/10.1145/344779.345009.
The concept of multipoint statistical simulation was introduced by Guardiano and Srivastava (1993). The significance of multipoint statistical simulation methods in relation to two-point statistical simulation methods is to enable generation of geological models comprising geological objects of complex shape, notably curvilinear. The principle of these methods provides information on the shape of the objects, their spatial distribution, and the way they are connected through the agency of a training image. This image can be a satellite image, a realization of another simulation method (an object method or a genetic process for example), a high-resolution seismic map, etc. A window that goes through the training image in order to capture the occurrence of the value configurations present is then defined. The training image thus acts as a substitute for the variogram which the two-point simulation methods refer to. Knowing the training image, it is desired to simulate a new image reproducing the shapes detected in the training image. The multipoint statistical simulation mode is sequential and the cells of the image to be simulated are visited one after the other in random order. From a practical point of view, it was not until Strebelle's teaching (2002) that a first algorithm, efficient enough to be usable, was implemented.
In parallel, analogous techniques known as texture synthesis methods have been developed in the field of computer graphics. Again, the principle of these methods refers to using a computer to provide a training image comprising patterns that are attempted to be reproduced in the simulated image. The information provided by this training image is extracted by a window that is slid over the image. This image synthesis is very rich and much work has been reported in the literature (for example (Efros and Leung, 1999) or (Wei and Levoy, 2000)). The cell visiting order in the case of texture synthesis is not random. For example, in the case of Efros and Leung (1999), the simulation process starts with the cell in the middle of the image to be simulated. Its closest neighbours are then sought, and then the closest neighbours of the closest neighbours, etc. A sequence of increasingly large concentric crowns is then obtained. In the case of Wei and Levoy (2000), the authors recommend that the cells of the image be visited in a predetermined order: from left to right and from bottom to top.
In order to be as representative of reality as possible and to allow optimum development of a reservoir, the construction of geological models must be able to account for any available external data such as, for example, measurements on cores extracted from wells or logging data.
Texture synthesis methods do not allow external data to be taken into account. Due to the absence of external data, the cells can be visited in a predetermined order. By visiting the cells close to those already containing values, the lateral extension of the objects is better preserved. Besides, smaller windows can be considered, which implies calculation time gains.
Multipoint statistical simulation methods allow external data to be taken into account in which event they are then referred to as conditional. In order to take into account of the conditioning points, they involve a random visiting order for the grid cells. In order to correctly reproduce large-size structures, this random visiting order requires consideration of windows of very large size, which has a time-consuming calculation. An effect referred to as “pixellation” (lack of continuity, from one cell to the next, of the physical property represented) can also sometimes be observed. The methods described in U.S. Published Patent Applications 2013/0,110,484 and 2011/0,251,833 introduce for the first time in the field of multipoint statistical simulation a possibility not necessarily requiring random path. The specification of a unilateral path is disclosed in U.S. Published Application 2011/0,251,833. However, although this path is mentioned, the path construction technique is not explained and its advantages are not shown.